Quantitative Economics, Volume 3, Issue 2 (July 2012)

Finding all pure-strategy equilibria in games with continuous strategies

Kenneth L. Judd, Philipp Renner, Karl Schmedders

Abstract

Static and dynamic games are important tools for the analysis of strategic interactionsamong economic agents and have found many applications in economics.In many games, equilibria can be described as solutions of polynomial equations.In this paper, we describe state-of-the-art techniques for finding all solutions ofpolynomial systems of equations, and illustrate these techniques by computing allequilibria of both static and dynamic games with continuous strategies.We computethe equilibrium manifold for a Bertrand pricing game in which the numberof equilibria changes with the market size.Moreover,we apply these techniques totwo stochastic dynamic games of industry competition and check for equilibriumuniqueness.Keywords. Polynomial equations, multiple equilibria, Bertrand game, dynamicgames,Markov-perfect equilibria.JEL classification. C63, C73, L13.